A simple 2D bio®lm model yields a variety of morphological
features
Slawomir W. Hermanowicz
*Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, USA
Received 10 April 2000; received in revised form 29 August 2000; accepted 10 October 2000
Abstract
A two-dimensional bio®lm model was developed based on the concept of cellular automata. Three simple, generic processes were included in the model: cell growth, internal and external mass transport and cell detachment (erosion). The model generated a diverse range of bio®lm morphologies (from dense layers to open, mushroom-like forms) similar to those observed in real bio®lm systems. Bulk nutrient concen-tration and external mass transfer resistance had a large in¯uence on the bio®lm structure. Ó 2001 Elsevier
Science Inc. All rights reserved.
Keywords:Bio®lm model; Bio®lm morphology; Cellular automata; External mass transfer; Erosion detachment
1. Introduction: bio®lms and their models
Bio®lms are a common form of microbial ecosystems associated with surfaces. They are found in extremely varied environments from `pure' water systems to stream beds, ship hulls and teeth surface. In response to varying environmental conditions, bio®lms develop dierent structures expressed in various morphologies. The richness of morphological forms has been recognized by many researchers (e.g., [1±7]). However, characterization of bio®lm morphology has been pri-marily restricted to qualitative descriptors such as `smooth', `fuzzy', `mushroomlike'. Recently, quantitative descriptors have been proposed including porosity and its gradients [8], connected porosity [9] and fractal dimensions [9±11]. These attempts are still in the developmental stage and it is likely that bio®lm morphology can be quanti®ed by more than one parameter. Better un-derstanding of bio®lm morphology is important not only for its characterization but for de-scription of mass transfer inside and around bio®lms. While experimental methods will ultimately www.elsevier.com/locate/mbs
*Tel.: +1-510 642 0151; fax: +1-510 642 7483.
E-mail address:hermanowicz@ce.berkeley.edu (S.W. Hermanowicz).
reveal bio®lm structure, mathematical models can be useful tools for investigation of the eects of dierent environmental conditions on bio®lm development and its morphology.
Early mathematical models of bio®lms were derived from similar models used in chemical en-gineering to describe diusion and reaction in a porous catalyst particle (e.g., [12]). Bio®lm was treated as a homogeneous matrix with uniformly distributed biochemical reactive sites. A nutrient penetrated through the matrix by molecular diusion and was transformed into products ac-cording to a prescribed local, intrinsic reaction rate. Such models were developed initially for one-dimensional geometry implying a ¯at bio®lm. Later, two-one-dimensional and three-one-dimensional models were introduced (for a recent mini-review, see Ref. [13]). Additional features were further incorporated such as multiple nutrients, multiple microbial species, variable bio®lm density [14± 20]. A majority of existing models assumed that the bio®lm composition and thickness were constant and described only nutrient transport and transformations. Some newer models included bio®lm development (growth, detachment) most commonly through a biomass displacement ve-locity that depended on the local microbial growth rate (see Ref. [17]). All these models, however, treated the bio®lm as a continuum and were based on dierential mass balances of various bio®lm components. More important, bio®lm morphology was prescribed in the models. In some models, bio®lm morphology was explicitly stipulated (e.g., as a ¯at layer) while in other models speci®c assumptions were made about bio®lm development (e.g., that the biomass displacement velocity is perpendicular to the substratum). In either case, the models were unable to describe complicated bio®lm morphologies observed experimentally in many systems.
We propose that a new class of bio®lm models is needed to overcome this de®ciency. Recently such a new class based on the idea of cellular automata has been presented [21±23]. All these in-dependently developed models subscribe to a similar philosophy but are dierent in details. In our work [22] we developed a simple model incorporating only an absolute minimum of assumptions necessary to describe realistically bio®lm development. This minimalistic approach is founded on two premises. First, we postulate that only processes of bio®lm development included in a model are those that are strictly necessary to yield realistic model outcomes, given the current state of knowledge. Second, modeler's preconceived notions about bio®lm organization should be kept to a minimum. Such limitations are achieved better if the rules of bio®lm development are local, i.e., they are formulated for a smallest set of model elements. If a model, which includes only a small set of features, adequately represents the reality of bio®lm structure, we are more convinced that these features are universal and common to generic bio®lms. While complex models might perform better in describing individual cases, they require additional parameters to achieve better ®t. Then, they often lose predictive powers as the choice of appropriate coecients becomes bewildering. Similar approach was advocated by Schweitzer [24]. He stated that for a model ``it is important to ®nd a level of description which on one hand considers the speci®c features of the system and re¯ects the origination of new qualities, but on the other hand is not ¯ooded with microscopic details''. From this point of view, cellular automata provide a suitable framework for model development.
2. Modeling aspects: cellular automata as a bio®lm model
set. The state of the object is changed depending on its value and inputs of other connected objects according to prescribed rules, `a transfer function'. Wolfram [25] presented a comprehensive description of cellular automata and their applications to many problems in science. In micro-biology, the development of microbial colonies on agar colonies was previously described with cellular automata [26].
To serve our stated purpose, the model must be robust since the detailed description of mi-crobial physiology in bio®lms is not yet fully developed. However, certain common features of bio®lm organization and development can be considered as universal, at least in a broad semi-quantitative sense. These features include:
· nutrient uptake and biomass growth;
· nutrient mass transfer inside and outside a bio®lm; · detachment of bio®lm fragments.
The objective of this work is to show how these general features can lead to the development of diverse bio®lm morphology. In particular, a set of simple development rules de®ned on a small scale (possibly as small as a single cell) results in the formation of self-organized structures.
2.1. Simple developmental rules
In the present work, the bio®lm consists of discrete units (`grid cells') which are embedded in a two-dimensional grid. The grid is typically rectangular and constitutes a working space in which the modeled bio®lm is allowed to develop. The states of each grid cell are zero or unity, corre-sponding to cells ®lled with water or biomass, respectively (Fig. 1). To avoid any confusion in the terminology, we will use the term `grid cell' to represent a model unit and `microbial cell' to refer to a real, living organism. The size of the grid cellsxis an arbitrary parameter based primarily on the desired model spatial accuracy. Since all features of the simulated bio®lm are related to the chosen cell size, the simulated structures can be simply scaled up or down through an appropriate value ofx. Thus, the resulting simulated bio®lms are scale-free and geometrically self-similar (in a
statistical sense). Hermanowicz and co-workers [9,10] found dierent morphological structures of a mixed-population, heterotrophic bio®lm depending on their geometric scales. Small structures (less than approximately 5 lm) were very compact as characterized by their fractal dimension
close to 2 (for 2-D cross-sections). Larger structures (with size above 10 lm) were much more
open and possibly self-similar. Their `fractal' dimension varied from 1.5 to 1.8 depending of the bio®lm age, distance from the substratum and overall water ¯ow direction. At present, it is not known whether these features are more commonly found in bio®lms or what is the upper cuto of the self-similarity. This type of analysis, however, could indicate an approximate size of smallest cell aggregates in a bio®lm and suggest a suitable value of the cell model size x.
In our model, a matrix representing nutrient concentrations is superimposed on the working space containing water and biomass. In this work, only a single limiting nutrient was considered. Thus, the nutrient concentration matrix has the same dimensions as the working space grid. Bio®lm development occurs in discrete time steps. At every time step, each occupied grid cell may divide with the probability Pd. In general, Pd can be a function of nutrient concentrations, cell metabolic status or other factors. In the present model, a simple Monod-like function was used to relate the probability Pd and the local limiting nutrient concentration c:
Pd c
cK: 1
Thus, at low nutrient concentrations, the grid cells will divide less frequently than at high nutrient concentrations. After each time step, the nutrient matrix is updated to describe a new nutrient concentration ®eld. The updated concentration ®eld re¯ects changes of nutrient uptake caused by new bio®lm geometry. Nutrient concentration gradients result from a combination of nutrient uptake by the biomass and nutrient transport from the bulk ¯uid through a concen-tration boundary layer and inside the bio®lm. The thickness of the concenconcen-tration boundary layer adjacent to the biomass/water interface characterizes the external mass transport. In general, it depends on the ¯ow regime. Zhang and Bishop [8] and DeBeer and co-workers [27] used microelectrodes to determine the nutrient concentration gradients and consequently the boundary layer thickness. DeBeer and co-workers [27] clearly demonstrated that the boundary layer thickness depended on the bulk water velocity. Yang and Lewandowski [28] also showed that mass transport outside the biomass clusters is much larger than inside the clusters. Pre-sumably outside the clusters, advection dominates the transport while diusion is the controlling factor inside the clusters.
In the model, a relatively simple representation of these processes was adopted. At each time step, (and thus at each step in bio®lm development) a concentration boundary layer is imposed at the bio®lm surface, i.e., for each occupied grid cell with at least one empty neighbor. The thickness of the boundary layer dB is a model parameter and is expressed in the dimensionless form as
dB dB=x. It is assumed that mass transport through the boundary layer and inside the biomass
(i.e., occupied grid cells) occurs by diusion with the eective diusion coecient D (same for biomass and water). Outside the boundary layer, a constant nutrient concentration cS is main-tained (representing advection). In such a setup, a steady state concentration ®eld can be de-scribed by a two-dimensional Poisson equation
wherer crepresents the intrinsic nutrient uptake rate. The appropriate boundary conditions are
where n is the direction perpendicular to the substratum S. Solving Eq. (2) with complicated geometry requires a lot of computational resources. In addition, Eqs. (2)±(4) still represent an idealization of a real bio®lm. In the light of the previously expressed modeling philosophy, it seems that a simpler approach to calculate nutrient concentration distribution could be used as long as it describes a reasonably similar pattern of concentration gradients. This approach is quite dierent from the work of Picioreanu and co-workers [23] who solved numerically the full Poisson equation. For a ¯at, one-dimensional bio®lm (and zero-order nutrient uptake kinetics) a con-centration pro®le through a bio®lm can be calculated by solving the one-dimensional version of Eq. (2) in an explicit form
where k is the uptake rate and d is the distance from the surface of the bio®lm (penetration distance). In an analogy with an electric circuit, one can consider that nutrient depletion is similar to a voltage drop across a resistor. The `resistance' to mass transport is in this case a function of the penetration distance d. In a two-dimensional bio®lm, nutrients can be supplied to a given point inside the bio®lm in many directions from the bio®lm surface. Again, in an analogy with resistor networks, the overall `resistance' to mass transfer was approximated as a harmonic mean of `resistances' in dierent directions. In the two-dimensional model, eight directions of nutrient supply were considered for each point inside the bio®lm and the nutrient concentration was calculated from a 2-D analog of Eq. (5)
c c1=2
wherediare penetration distances (see Fig. 1). Thus, at each time step, eight penetration depthsdi were calculated for each occupied grid cell and the corresponding nutrient concentration c was found from Eq. (6). Then, the probability of division for each occupied grid cell was computed from Eq. (1). The validity of this approach is assessed in Section 3.
rule if no empty space is available. Obviously, this rule is very simplistic and does not completely describe the behavior of a real bio®lm. However, mechanisms governing displacement of dividing microbial cells in a real bio®lm are not known. They probably include a combination of increasing local microbial cell density and displacement of bio®lm components. Nonetheless, the proposed model rule might be a reasonable ®rst step approximation. If mechanisms of microbial cell dis-placement in a real bio®lm become known better, the rule may be modi®ed and its eects on bio®lm simulation reevaluated. In this capacity, a plausible bio®lm model might be useful not only for formulating a speci®c hypothesis but also for its testing.
In addition to biomass growth and displacement, parts of the bio®lm can detach from the rest of the matrix. Although this process appears to be very important for bio®lm development, little is known about the phenomena which control bio®lm detachment. Chang et al. [29], Peyton and Characklis [30], and Gjaltema and co-workers [31] reported bio®lm detachment kinetics deter-mined through multivariate regression analysis of several operational parameters. These corre-lations are of limited value for model development since they describe only case-speci®c average rates and not local detachment events. Other cell-based models [21,23] did not include detach-ment. Lacking any detailed information on local bio®lm detachment, a very simple rule was again adopted in the present model. For grid cells at the biomass/liquid interface (i.e., for grid cells with at least one empty neighbor) the probability of cell erosion should be an increasing function of the hydrodynamic shear stress s. In addition, the `strength' of bio®lm (its cohesion) should be also involved. We proposed to characterize this poorly de®ned feature with a single parameterr. Thus, with an increasing bio®lm strength r, the probability of erosion should decrease. To accommo-date this qualitative but intuitive description, the probability of cell erosion Pe was described by the following simple function:
Pe 1
1 r=s: 7
The probability Pe is shown in Fig. 2 as a function of r and s. Because the bio®lm strength r appears only as a ratio to the shear stresss, there is no need for its physical interpretation. Simply, when r=s1, the surface grid cells detach with the probability 1/2.
If, as a result of cell erosion, a cluster of cells became completely detached from the rest of the bio®lm or the substratum, the whole cluster is eliminated from the work space. Such an erosion mechanism allowed for detachment of not only individual cells but also of larger clusters if their connections with the rest of the biomass of the substratum were severed. This feature was espe-cially important for dendritic, rami®ed bio®lm morphologies (see Fig. 4) where large clusters were often attached to the substratum with a narrow `neck'.
could be estimated from the following considerations. If for all occupied cells cK, then the probability of division Pd is close to 1 (see Eq. (1)). Therefore, at each time step, all cells would divide and the biomass would double. It follows that the duration of a time step is comparable with the minimum doubling timetminln 2=l
max.
3. Computational methods
The developed model was implemented as a computer code using IDL language (RSI, Boulder, CO, USA) on a Windows 95 and NT platforms. The IDL language oers a few convenient features such as erosion and dilation functions which were used to create the boundary layer. For example, the boundary layer adjacent to a complex biomass/liquid interface (i.e., interface be-tween occupied and empty grid cells) was created by dilating the set of occupied cells by the speci®ed width. The dilation of the biomass was the result of a convolution of the working space matrix with a one dimensional row- or column-vector as a kernel. The length of the vector de-termined the thickness of the boundary layer. Erosion of the biomass is a dual function to the dilation of the void (unoccupied) space. The basis of the IDL procedures were described in [32]. Similarly, the interface itself could be determined by subtracting from the set of the occupied cells the same set eroded by one cell width. The connectivity function, also available in IDL, allowed for a convenient determination of cell clusters detached through the erosion of a single cell. Al-though IDL oered a convenient tool for model coding, any other suitable language can be used. In the model, a simple analytical equation (Eq. (6)) was used to calculate the concentration in and around the bio®lm. This approach was adopted because it simpli®ed the program structure and dramatically decreased computational eort. The results were compared with the numerical solution of the full Poisson equation (Eq. (2)) which was solved on the same grid using a relax-ation method. In this method, 10 000 steps were used to eliminate the in¯uence of initial
tions. The results are shown in Fig. 3 forF 0:01 andCS1 for an irregular bio®lm cluster. No ¯ux boundary conditions were assumed at the substratum (top of the grid) and constantCS1 for three remaining grid sides. Although the numerical solution was not identical with the sim-pli®ed analytical equation, they coincided quite closely, certainly within any experimental accu-racy.
Simulations were performed for grid sizes from 5050 to 200200 cells. Each simulation was terminated either after a speci®ed number of steps (typically 50±100) or if the occupied cells ®lled more then 80% of the working space. The simulations were carried out for the following values of model parameters:F 0:01, 0.05, 0.1, 0.5;CS1:5, 3, 5, 7, 9, 11, 13, 15;dB1, 3, 5, 7, 9, 11 and
r=s0:1, 0.5, 1, 5. Three repetitions were performed for each set of parameters yielding a total of 1152 simulations. For each of the simulated bio®lms several morphological characteristics were determined including its porosity, surface extension and nutrient ¯ux. Since the bio®lm in each simulation grew to a dierent thickness, the porosityewas de®ned as the fraction of unoccupied cells in a rectangle of the smallest heightHmin wholly containing the bio®lm
e1ÿ Noccupied
LsubstrHmin; 8
whereNoccupiedis the number of the occupied grid cells in the work space, andLsubstris the length of the substratum in grid cell units. Internal porosity of the bio®lmeI was de®ned as the fraction of
unoccupied grid cells without connection to the bulk liquid. These cells were also counted in the smallest rectangle embedding the bio®lm. Internal porosityeIis a measure of `holes' in the bio®lm
which remain un®lled because of growth restrictions due to mass transfer limitations of the nu-trient supply.
Picioreanu et al. [23] used surface enlargement SE as a measure to quantify bio®lm morphol-ogy. In our work, we de®ned SE similarly as
SELinterface
Lsubstr ; 9
whereLinterface is the length (in cell units) of the interface between occupied and empty grid cells. Typically, SE>1 (sometimes SE1) due to irregular and convoluted bio®lm interface. How-ever, when the simulated bio®lm was patchy and covered only a part of the substratum, SE could be less than 1.
4. Results and discussion
In our work, only four parameters, CS;dB;r=s and F were needed to describe bio®lm devel-opment. Two of these, the dimensionless bulk nutrient concentration CS and the dimensionless thickness of the boundary layer dB, can be controlled or easily monitored in a reactor thus
ex-erting an external in¯uence on bio®lm development. Preliminary results reported earlier [33] suggested that the external mass transfer characterized by the boundary layer thickness greatly aected the morphology of the simulated bio®lm. In this work, the eects ofdB together withCS,
and r=s were further investigated. Fig. 4 displays one of the simulation results (for F 0:01;CS2:6;dB5 andr=s0:2). This ®gure shows the values of nutrient concentrations C, probability of divisionPd and the cell age (i.e., the time step in which each cell was created by division). A concentration gradient is established through the boundary layer and the bio®lm. As a result of these mass transfer limitations, cell divisions are predominantly in the outer zone of the bio®lm. On the right side of the bio®lm cluster a channel is open to advection since its width
exceeds 2dB. On the left, only diusional mass transport is possible because the channel width is smaller than 2dB.
Results of more simulations are shown in Fig. 5 which presents examples of bio®lm mor-phology developed at dierent values of CS, dB and r=s. These results are further quanti®ed through surface enlargement SE, bio®lm porosity e, and inner porosity eI. Fig. 6 shows the
evolution of surface enlargement SE in response to model parameters. Each of the six panels in Fig. 6 presents the changes of the evaluated variable (SE) in response toCSfor the speci®ed value of dB. Similarly, Fig. 7 shows the variations of bio®lm porosity e, and Fig. 8 presents the inner
porosity eI.
As seen in Fig. 5, the model was capable of producing a wide range of bio®lm morphologies. These diverse structures were generated solely in response to changes of model parameters and not explicitly speci®ed. Thus, the modeling results suggest that a few simple and generic processes in bio®lm development can be responsible for greatly diversi®ed bio®lm structures observed in na-ture. An increase in the nutrient concentrationCSaects bio®lm structure in a complex fashion. It is noteworthy that a small change ofCScan yield a very big dierence in bio®lm development. At low values of CS, very little bio®lm growth occurred. At best, only a few cells hugged the sub-stratum while most growing cells were removed by erosion. For example, atdB1 andr=s0:5
(bottom left panel in Fig. 5) virtually no bio®lm developed at CS1:5. Initially deposited cells were unable to grow faster than they were removed from the substratum by erosion. Yet, a small increase of the nutrient concentration to CS 3, produced a dense layer of biomass. This major shift, reminiscent of a phase transition, is also seen in Fig. 6 where SE changes from almost 0 to approximately 2. Bio®lm porosity also increased with the initial increase of CS (Fig. 7). Bio®lm strength (as represented by r=s) shifted the transition towards lower nutrient concentrations
Fig. 6. Surface extension SE and examples of bio®lm morphology forF0:05.
(compare the top and bottom rows in the bottom right panel in Fig. 5) but otherwise had little eect.
The transition from virtually no bio®lm to an abundant bio®lm was most pronounced at lowdB
(i.e., at low external mass transfer resistance). As dB was increased, the transition became more
gradual with intermediate forms of bio®lm appearing. Comparing bio®lm morphology evolution at dB 5 (middle left panel in Fig. 5) with dB1 (lower left panel in Fig. 5) illustrates this
dierence. Similarly, Fig. 6 shows a more gradual increase of SE at higherdB. AsCScontinued to
increase, both SE and e reached a maximum and then decreased. At the highest nutrient con-centrations, the bio®lm became less porous and more compact due to increased nutrient pene-tration promoting growth inside the bio®lm. Similarly, the highest inner porosity eI (Fig. 8) is
realized at intermittent values of CS when nutrient concentration is high enough for bio®lm growth but, at the same time, not high enough to penetrate throughout the structure and supply nutrients to the inner holes. The concentrationCSat which bio®lm porosityereached a maximum, shifted towards higher values for larger dB.
The changes ofdB also have other eects on bio®lm porositye (Fig. 7). At small dB (i.e., for
small external mass transfer resistance), densely layered bio®lms were developed. AsdBincreased,
the bio®lm developed in a more open, dendritic form. This phenomenon is typical to many dif-fusion-limited growth or aggregation processes (see [34]). Growth occurred primarily in the outer parts of the bio®lms while the inner parts were shielded from nutrient supply and growth. These shielding eects of the boundary layer were less evident at higher nutrient concentrations CS because more nutrient was available for growth even at higher mass transfer resistance.
Typically, the SE had a weak maximum in response to the variations of dB. At low dB, the
increasing dB, the SE also increased as a result of the branching growth discussed previously.
However, at very largedB the shielding eect resulted in the growth of individual `trees' separated
one from another. (Compare bio®lm morphology dB 5 with that at dB 11.) Although each
`tree' was highly branched, the overall value of the interface area decreased due to their separation.
5. Conclusions
A bio®lm model was developed using the cellular automata approach. The proposed model describes only the simplest case of a single-species bio®lm with a single growth-limiting nutrient but the developed framework can easily be used for more complex modeling tasks. The results of modeling suggested that a small change of the bulk nutrient concentration can have a dramatic eect on the bio®lm development. At the concentrations below the critical value, almost no bio®lm can develop while at slightly higher concentrations, bio®lm development is bountiful. This `phase-like' transition is especially sharp for thin external boundary layers. Bio®lm porosity and interfacial surface reach a maximum at an intermediate nutrient concentration. The value of this concentration tends to increase with thicker boundary layers.
The thickness of the boundary layer has a very strong in¯uence on bio®lm morphology. Thin boundary layers (hence low mass transfer resistance) promote the growth of dense and compact bio®lms. At thicker boundary layers, much more open, dendritic bio®lm forms develop, resem-bling `mushrooms' or `tulips' observed in some real systems. The eect of bio®lm strength or cohesion have minor eects on its development.
The proposed model is capable not only of simulating a wide range of distinct bio®lm mor-phologies but also suggesting that such diversity can possibly result from self-organization of small, individual units. The units evolve in accordance with a small set of minimally de®ned, local rules identical for all units. The dierences in generated structures are not externally imposed but result from changes of very basic characteristics of the environment. Thus, the model can be a useful tool for the analysis of bio®lm development.
References
[1] H. Siegrist, W. Gujer, Mass transfer mechanism in a heterotrophic bio®lm, Water Res. 19 (1985) 1369.
[2] F.R. Christensen, G.H. Kristensen, J.L.C. Jansen, Bio®lm structure ± an important and neglected parameter in wastewater treatment, Water Sci. Technol. 21 (1989) 805.
[3] J.R. Lawrence, D.R. Korber, B.D. Hoyle, J.W. Costerton, D.E. Caldwell, Optical sectioning of microbial bio®lms, J. Bacteriol. 173 (1991) 6558.
[4] S. Kugaprasatham, H. Nagaoka, S. Ohgaki, Eects of turbulence on nitrifying bio®lms at nonlimiting substrate conditions, Water Res. 26 (1992) 1629.
[5] D. DeBeer, P. Stoodley, Z. Lewandowski, Liquid ¯ow in heterogeneous bio®lms, Biotechnol. Bioeng. 44 (1994) 636.
[6] A.A. Massoldeya, J. Whallon, R.F. Hickey, J.M. Tiedje, Channel structures in aerobic bio®lms of ®xed®lm reactors treating contaminated groundwater, Appl. Environ. Microbiol. 61 (1995) 769.
[7] P.S. Stewart, R. Murga, R. Srinivasan, D. DeBeer, Bio®lm structural heterogeneity visualized by three microscopic methods, Water Res. 29 (1995) 2006.
[9] S.W. Hermanowicz, U. Schindler, P. Wilderer, Fractal structure of bio®lms: new tools for investigation of morphology, Water Sci. Technol. 32 (1995) 99.
[10] S.W. Hermanowicz, U. Schindler, P. Wilderer, Anisotropic morphology and fractal dimensions of bio®lms, Water Res. 30 (1996) 753.
[11] W. Zahid, J. Ganczarczyk, Fractal properties of the RBC bio®lm structure, Water Sci. Technol. 29 (1994) 217. [12] P. Harremoes, Bio®lms Kinetics. Water Pollution Microbiology, vol. 2, Wiley, New York, 1978, p. 71.
[13] D.R. Noguera, S. Okabe, C. Picioreanu, Bio®lm modeling: present status and future directions, Water Sci. Technol. 39 (1999) 273.
[14] O. Wanner, W. Gujer, Multiple species bio®lm model, Biotechnol. Bioeng. 28 (1986) 314.
[15] H. Furumai, B.E. Rittmann, Advanced modeling of mixed populations of heterotrophs and nitri®ers considering the formation and exchange of soluble microbial products, Water Sci. Technol. 26 (1992) 493.
[16] B.E. Rittmann, J.A. Manem, Development and experimental evaluation of a steady-state multispecies bio®lm model, Biotechnol. Bioeng. 39 (1992) 914.
[17] O. Wanner, Modeling of mixed-population bio®lm accumulation, in: G.G. Geesey, Z. Lewandowski, H.C. Flemming (Eds.), Biofouling and Biocorrosion in Industrial Water Systems, Lewis, Boca Raton, FL, 1994, p. 37. [18] O. Wanner, New experimental ®ndings and bio®lm modelling concepts, Water Sci. Technol. 32 (1995) 133. [19] O. Wanner, P. Reichert, Mathematical modeling of mixed-culture bio®lms, Biotechnol. Bioeng. 49 (1996) 172. [20] D.R. Noguera, G. Pizarro, D. Stahl, B.E. Rittmann, Simulation of multispecies bio®lm development in three
dimensions, Water Sci. Technol. 39 (1999) 123.
[21] J.W.T. Wimpenny, R. Colasanti, A unifying hypothesis for the structure of microbial bio®lms based on cellular automaton models, FEMS Microbiol. Ecol. 22 (1997) 1.
[22] S.W. Hermanowicz, A model of two-dimensional bio®lm morphology, in: Proceedings of the Second International Conference on Microorganisms in Activated Sludge and Bio®lm Processes, University of California, Berkeley, CA, 21±23 July 1997; Water Sci. Technol. 37 (1997) 219.
[23] C. Picioreanu, M.C.M. Van Loosdrecht, J.J. Heijnen, Mathematical modeling of bio®lm structure with a hybrid dierential-discrete cellular automaton approach, Biotechnol. Bioeng. 58 (1998) 101.
[24] F. Schweitzer, Self-organization of complex structures from individual to collective dynamics ± some introductory remarks, in: F. Schweitzer (Ed.), Self-organization of Complex Structures from Individual to Collective Dynamics, Gordon and Breach, London, 1997, p. xix.
[25] S. Wolfram, Theory and Applications of Cellular Automata, World Scienti®c, Singapore, 1986.
[26] E. Ben-Jacob, I. Cohen, Adaptive self-organization of bacterial colonies, in: F. Schweitzer (Ed.), Self-organization of Complex Structures from Individual to Collective Dynamics, Gordon and Breach, London, 1997, p. 243. [27] D. DeBeer, P. Stoodley, P.Z. Lewandowski, Liquid ¯ow and mass transport in heterogeneous bio®lms, Water Res.
30 (1996) 2761.
[28] S. Yang, Z. Lewandowski, Measurement of local mass transfer coecient in bio®lms, Biotechnol. Bioeng. 48 (1995) 737.
[29] H.T. Chang, B.E. Rittmann, D. Amar, R. Heim, Bio®lm detachment mechanisms in a liquid-¯uidized bed, Biotechnol. Bioeng. 38 (1991) 499.
[30] B.M. Peyton, W.G. Characklis, A statistical analysis of the eect of substrate utilization and shear stress on kinetics of bio®lm detachment, Biotechnol. Bioeng. 41 (1993) 728.
[31] A. Gjaltema, L. Tijhuis, M.C.M. van Loosdrecht, J.J. Heijnen, Detachment of biomass from suspended nongrowing spherical bio®lms in airlift reactors, Biotechnol. Bioeng. 46 (1995) 258.
[32] R.M. Haralick, S.R. Sternberg, X.H. Zhuang, Image-analysis using mathematical morphology, IEEE Trans. Pattern Anal. Machine Intell. 9 (1987) 532.
